There seems to be conflicting information on the internet.
Are super capacitors polarized? For instance, could you charge them and then suddenly invert their polarity (à la charge pump)?
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Supercapacitors are polarized and must operate with the correct polarity. Polarity is controlled by design with asymmetric electrodes, or, for symmetric electrodes, by a potential applied during manufacture.
Although the anode and cathode of symmetric supercapacitors consist of the same material, theoretically supercapacitors have no true polarity. Normally catastrophic failure does not occur, however reverse-charging a supercapacitor lowers its capacity. It is recommended practice to maintain the polarity resulting from a formation of the electrodes during production. Asymmetric supercapacitors are inherently polar.
Supercapacitors may not be operated with reverse polarity, precluding AC operation.
In other words, even for capacitors with symmetric electrode structures, there is a "charging" process during the manufacture that creates a asymmetric insulating layer inside the capacitor. You can reverse-charge a super capacitor, but doing so likely leads to a microscopic degradation of the specially prepared electrode surfaces (probably due to bizarre chemistry physics I don't understand), reducing the available capacitance.
I suspect this is similar to the process for manufacturing lead-acid batteries, where the plates are actually initially identical, and the initial charging process modifies the chemical properties of both plates. In their un-conditioned state, it's probably possible to electrochemically convert either electrode to the "positive" electrode, but once the electrode have been conditioned, trying to reverse their polarization would probably damage their chemical properties, and thus the capacitor.